What is Bayesian Statistics?

What is Bayesian Statistics?

Bayesian statistics is a branch of statistics that is based on the Bayesian probability theory. It provides a framework for updating beliefs or probabilities about a hypothesis as new evidence or data becomes available. In contrast to classical (frequentist) statistics, which treats probabilities as frequencies or limiting proportions, Bayesian statistics views probabilities as measures of belief or certainty.

Bayesian Probability

Probability in the Bayesian context reflects the degree of belief or certainty about the occurrence of an event. It is updated as new information is acquired, incorporating prior knowledge and observed data through Bayes' theorem.

Prior Probability

The initial probability assigned to a hypothesis before considering any new evidence. It represents existing knowledge or beliefs about the likelihood of an event before incorporating data.

Likelihood

The probability of observing the given data, given a particular hypothesis. It describes the compatibility between the observed data and the hypothesis.

Posterior Probability

The updated probability of a hypothesis after taking into account both prior knowledge and new evidence. It is calculated using Bayes' theorem, combining the prior probability and the likelihood.

Bayes' Theorem

A fundamental formula in Bayesian statistics that calculates the posterior probability of a hypothesis given prior knowledge and observed data. It is expressed as P(H|D) = P(D|H) * P(H) / P(D), where P(H|D) is the posterior probability, P(D|H) is the likelihood, P(H) is the prior probability, and P(D) is the probability of the observed data.

Posterior Distribution

The probability distribution of the parameter(s) of interest after incorporating prior knowledge and observed data. It represents the updated beliefs about the parameter(s).

Prior Distribution

The probability distribution representing the initial beliefs about the parameter(s) before observing any data. It is based on existing knowledge or subjective assessments.

Conjugate Prior

In Bayesian statistics, a prior distribution which belongs to the same family of probability distributions as the posterior distribution when combined with a specific likelihood function. Conjugate priors simplify calculations and result in closed-form solutions.

Markov Chain Monte Carlo (MCMC)

A computational technique widely used in Bayesian statistics to approximate the posterior distribution of parameters. MCMC methods, such as the Metropolis-Hastings algorithm and Gibbs sampling, are valuable for complex models where analytical solutions are challenging.

Bayesian Model Averaging

A technique in Bayesian statistics that considers multiple models and their associated parameter values to make predictions or inferences. It accounts for model uncertainty by weighing the contribution of each model based on its posterior probability.

Bayesian Hypothesis Testing

A method of hypothesis testing within the Bayesian framework, where the focus is on updating beliefs about the relative plausibility of different hypotheses given the observed data.

Subjective Bayesianism

An approach to Bayesian statistics that emphasizes the incorporation of subjective beliefs and opinions into the analysis. It recognizes that prior probabilities may be subjective and can be based on individual or expert judgment.